Tropical Abel-Jacobi theory
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912337548541952 |
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| author | Amini, Omid Corey, Daniel Monin, Leonid |
| author_facet | Amini, Omid Corey, Daniel Monin, Leonid |
| contents | To a compact tropical variety of arbitrary dimension, we associate a collection of intermediate Jacobians defined in terms of tropical homology and tropical monodromy. We then develop an Abel-Jacobi theory in the tropical setting by defining functorial Abel-Jacobi maps. We introduce, in particular, tropical Albanese varieties and formulate obstructions to algebraic equivalence of tropical cycles. In dimension 1, we show that this recovers the existing Abel-Jacobi theory for tropical curves.
As an application, we consider the Ceresa class of a tropical curve which is defined as the image of the Ceresa cycle in an appropriate intermediate Jacobian under the Abel-Jacobi map. We give an explicit formula for this class entirely in terms of the combinatorics of the tropical curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14415 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tropical Abel-Jacobi theory Amini, Omid Corey, Daniel Monin, Leonid Algebraic Geometry Combinatorics Number Theory Primary: 14T10, 14T20, Secondary: 05C25, 05E14, 14C25, 14H40 To a compact tropical variety of arbitrary dimension, we associate a collection of intermediate Jacobians defined in terms of tropical homology and tropical monodromy. We then develop an Abel-Jacobi theory in the tropical setting by defining functorial Abel-Jacobi maps. We introduce, in particular, tropical Albanese varieties and formulate obstructions to algebraic equivalence of tropical cycles. In dimension 1, we show that this recovers the existing Abel-Jacobi theory for tropical curves. As an application, we consider the Ceresa class of a tropical curve which is defined as the image of the Ceresa cycle in an appropriate intermediate Jacobian under the Abel-Jacobi map. We give an explicit formula for this class entirely in terms of the combinatorics of the tropical curve. |
| title | Tropical Abel-Jacobi theory |
| topic | Algebraic Geometry Combinatorics Number Theory Primary: 14T10, 14T20, Secondary: 05C25, 05E14, 14C25, 14H40 |
| url | https://arxiv.org/abs/2504.14415 |